Homework Ch 21b RC Switching: Charge and Discharge

Problem

The capacitor in the figure (assume all values are accurate to 3 significant figures) is initially uncharged and the switch, in position c, is not connected to either side of the circuit. The switch is now flipped to:

• position a for 10.0 ms,
• then to position b for 10.0 ms,
• and then brought back to position c.

What is the final potential difference across the capacitor?

Final answer: 1.84 V

• Solve the problem above by carefully following the instructions below.

Learning Objectives

• Compute the time constant = RC for an RC circuit.
• Use exponential charging: VC(t) = V0(1 – e-t/) (starting from 0 V).
• Use exponential discharging: VC(t) = Vinitial e-t/.
• Track the capacitor voltage through multiple time intervals and switching steps.
• Present a clear, organized solution with correct notation, units, and significant figures.

Note: If a quantity remains constant and therefore requires no calculation, clearly state this and briefly explain why it remains unchanged (e.g., at position c the capacitor is disconnected). Also note that initial voltage of one step is the final voltage of the previous step.

Grading Details (5 pts total)

The following steps must be shown. Missing them will mean missing points.

1. Circuit Interpretation + Diagram

• Draw the circuit in each switch position: a (charging), b (discharging).
• Clearly state the initial condition: VC(0) = 0.

1. General Formulae (Before Numbers)

• Write the time constant: = RC.
• Write the charging equation for step a with the correct battery voltage V.
• Write the discharging equation for step b starting from the capacitor voltage at the end of step a.

1. Symbolic Setup (Two-Step Process)

• Compute V1 = capacitor voltage after 10.0 ms in position a (symbolic first, then numbers).
• Compute V2 = capacitor voltage after the next 10.0 ms in position b (symbolic first, then numbers).
• State why VC,final = V2 once the switch returns to c.

1. Substitution and Mathematical Steps

• Use: R = 150 , C = 40 F, t = 10 ms, and V = 12 V.
• Show the exponential calculations clearly (do not jump directly to the final number).
• Keep units consistent (seconds, farads, ohms, volts).

1. Final Answer + Clarity

• No credit will be given for this step if steps 14 are skipped.
• Box or highlight the final capacitor voltage when the switch returns to position c.
• Report the result with correct unit and appropriate significant figures.
• The steps can be followed first for charging then for discharging.

Submission Guidelines

• Follow the page requirements (writing, color rules, single PDF, clarity, upload method).
• Your submission must show the complete solution to this problem only. Ensure the uploaded file clearly displays the entire solution, with no steps cut off by page margins or poor image quality.
• After you submit: Open Submission Details View Feedback to confirm the PDF is upright, readable, and properly framed. If not, re-submit before the due time.

Late Policy

• On-time: Submit by the Due Date No penalty.
• Grace period: Between the Due Date and Available Until 10% deduction.
• After Available Until: Assignment locks; no submissions accepted Score will be 0.
• Grading: Homework is manually graded. Canvas may show 0 points until grading is complete; youll receive a notification when finished.

Submission Quality Deductions (applied to total score)

• Orientation/Legibility issues (rotated, upside down, too small/large, or blurry): 0.5 point each
• Too small handwriting (to the point that makes it hard to read and/or requires the reader to zoom in): 0.5 point
• Files added in comments or email (must be one PDF via the assignment upload only): 1 point (still must be before due date)
• Color policy violations (e.g., red/pink used for work): 0.5 point
• Did not verify submission (obviously unviewable PDF that a quick check would have caught): 0.5 point
• Repeat Issues: If the same submission problems (e.g., orientation, file format, color policy, or verification issues) occur in future assignments, the deduction will increase by an additional 0.5 points for each repeated violation.
• Unreadable or can’t be reviewed: If your submission is completely unreadable or cannot reasonably be reviewed, the entire assignment may receive a score of 0.

All work and diagrams must be your own handwriting. Typed or copied solutions may be dismissed.